7.
5.3. GEOMETRIC REPRESENTATION OF C 95THEOREM 5.2.2 (Gauss) If f (z) = an z n + an−1 z n−1 + · · · + a1 z + a0 ,where an = 0 and n ≥ 1, then f (z) = 0 for some z ∈ C.It follows that in view of the factor theorem, which states that if a ∈ F isa root of a polynomial f (z) with coeﬃcients from a ﬁeld F , then z − a is afactor of f (z), that is f (z) = (z − a)g(z), where the coeﬃcients of g(z) alsobelong to F . By repeated application of this result, we can factorize anypolynomial with complex coeﬃcients into a product of linear factors withcomplex coeﬃcients: f (z) = an (z − z1 )(z − z2 ) · · · (z − zn ).There are available a number of computational algorithms for ﬁnding goodapproximations to the roots of a polynomial with complex coeﬃcients.5.3 Geometric representation of CComplex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). The representation is known as the Arganddiagram or complex plane. The real complex numbers lie on the x–axis,which is then called the real axis, while the imaginary numbers lie on they–axis, which is known as the imaginary axis. The complex numbers withpositive imaginary part lie in the upper half plane, while those with negativeimaginary part lie in the lower half plane. Because of the equation (x1 + iy1 ) + (x2 + iy2 ) = (x1 + x2 ) + i(y1 + y2 ),complex numbers add vectorially, using the parallellogram law. Similarly,the complex number z1 − z2 can be represented by the vector from (x2 , y2 )to (x1 , y1 ), where z1 = x1 + iy1 and z2 = x2 + iy2 . (See Figure 5.1.) The geometrical representation of complex numbers can be very usefulwhen complex number methods are used to investigate properties of trianglesand circles. It is very important in the branch of calculus known as ComplexFunction theory, where geometric methods play an important role. We mention that the line through two distinct points P1 = (x1 , y1 ) andP2 = (x2 , y2 ) has the form z = (1 − t)z1 + tz2 , t ∈ R, where z = x + iy isany point on the line and zi = xi + iyi , i = 1, 2. For the line has parametricequations x = (1 − t)x1 + tx2 , y = (1 − t)y1 + ty2and these can be combined into a single equation z = (1 − t)z1 + tz2 .